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Higher Derivatives01:29

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In calculus, higher-order derivatives extend the idea of differentiation beyond the first derivative to capture successive rates of change. These derivatives provide detailed information about the behavior of functions and have important applications in both mathematics and physics. To illustrate these concepts, consider the example functionwhich serves as a useful case study for exploring higher derivatives.The first derivative represents the slope of the original function. The second...
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The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
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DerivativesThe concept of instantaneous rate of change is fundamental in both mathematics and physics, particularly in describing how a moving object alters its position with respect to time. This rate is captured mathematically through the derivative of a function. The derivative at a point represents the slope of the tangent line to the curve of the function at that point and quantifies how the function’s output changes per infinitesimal change in input.Derivative of the Square Root...
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A ship tracking an approaching aircraft relies on geometric measurements to find out the aircraft’s position relative to the observer. By measuring the slant distance to the aircraft and the angle of elevation, the horizontal and vertical components of the distance can be obtained using trigonometric relationships. This geometric approach provides a basis for analyzing how the observed angle changes as the aircraft moves closer to the ship.To examine the mathematical behavior of the angle...
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Elliptical arches are fundamental in architectural and structural engineering, offering aesthetic appeal and structural efficiency. The shape of an elliptical arch follows a constrained geometric relationship where the height and horizontal position are implicitly related. This means that the height y cannot be explicitly expressed as a function of the horizontal position x, necessitating implicit differentiation for slope and curvature analysis.The equation of an ellipse centered at the origin...
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In calculus, the concept of antiderivatives serves as the reverse operation of differentiation, akin to retracing the steps of a dynamic process to determine its initial state.An antiderivative of a function f(x) is another function F(x) such that its derivative yields the original function:Since differentiation eliminates constant terms, an antiderivative is not unique; instead, it includes an arbitrary constant C, leading to the general form:This constant accounts for unknown initial...
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Related Experiment Video

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Deep Neural Networks for Image-Based Dietary Assessment
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Derivative Kernels: Numerics and Applications.

Mahdi S Hosseini, Konstantinos N Plataniotis

    IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
    |June 15, 2017
    PubMed
    Summary

    A new framework for numerical differentiation (ND) creates highly localized, ripple-free filters. This method enables accurate image gradient and Hessian approximation for various applications, enhancing image processing techniques.

    Area of Science:

    • Signal Processing
    • Image Analysis
    • Numerical Methods

    Background:

    • Numerical differentiation (ND) is crucial for image analysis.
    • Existing methods like Gaussian kernels have limitations in frequency localization and artifacts.

    Purpose of the Study:

    • To propose a generalized framework for constructing finite impulse response (FIR) filters for numerical differentiation.
    • To develop 2D MaxFlat kernels for precise image directional differentiation.

    Main Methods:

    • Utilizing interpolating power polynomials and maximally flat design techniques.
    • Constructing a library of 2D MaxFlat kernels for arbitrary derivative order, cutoff level, and steering angle.

    Main Results:

    • The proposed ND filter exhibits sharp localization in the Fourier domain with no ripple artifacts.

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  • Demonstrated accurate discrete approximation of gradients, Hessian, and higher-order tensors.

    • Conclusions:

    • The 2D MaxFlat kernel library offers a versatile and robust solution for image differentiation.
    • The framework shows high efficiency and parameter control accuracy in applications like unsharp masking.